Etale同伦型与双单超覆盖
Etale Homotopy Types and Bisimplicial Hypercovers
摘要：
研究了点局部连通小Grothendieck中心$(\mc{C}, x)$上任意点局部连通单层$(X, z)$所对应的同伦型$T(X, z)$.证明了这类$T(X, z)$是点连通格式$X$的Artin-Mazur的étale同伦类型的特例，它在点局部弱等价下直到亲同构是不变的（但早期的证明见\cite{Schmidt1}），它恢复了$X$的常系数交换层和非交换层上同调.本文将这类$T(X, z)$与由$x = (X, z)$的点双单超覆盖的对角线构造的同伦型$T_b(X, z)$在上圈范畴上进行了比较，证明了对于任何局部不变性的目标对象$y$，在路径分量水平上都存在双射π_0 H_{\hyp}（x，y）\cong π_0 H_{\bihyp}（x，y）.这很快导致$\Ho{\sSet_\ast}$中的自然亲同构$T(X, z) \cong T_b(X, z)$。通过推论，人们立即建立了这样一个事实，即$T_b(X, z)$在指向的局部弱等价下直到亲同构都是不变的。这些类型的未指向版本的类似语句也如下。
英文摘要：
An étale homotopy type $T(X, z)$ associated to any pointed locally fibrant connected simplicial sheaf $(X, z)$ on a pointed locally connected small Grothendieck site $(\mc{C}, x)$ is studied. It is shown that this type $T(X, z)$ specializes to the étale homotopy type of Artin-Mazur for pointed connected schemes $X$, that it is invariant up to pro-isomorphism under pointed local weak equivalences (but see \cite{Schmidt1} for an earlier proof), and that it recovers abelian and nonabelian sheaf cohomology of $X$ with constant coefficients. This type $T(X, z)$ is compared to the étale homotopy type $T_b(X, z)$ constructed by means of diagonals of pointed bisimplicial hypercovers of $x = (X, z)$ in terms of the associated categories of cocycles, and it is shown that there are bijections π_0 H_{\hyp}(x, y) \cong π_0 H_{\bihyp}(x, y) at the level of path components for any locally fibrant target object $y$. This quickly leads to natural pro-isomorphisms $T(X, z) \cong T_b(X, z)$ in $\Ho{\sSet_\ast}$. By consequence one immediately establishes the fact that $T_b(X, z)$ is invariant up to pro-isomorphism under pointed local weak equivalences. Analogous statements for the unpointed versions of these types also follow.
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